To determine which graph represents the given system of inequalities, we need to:
1. Understand the boundaries:
- Inequality 1: [tex]\( y \geq -5x + 2 \)[/tex]
- This translates to the boundary line [tex]\( y = -5x + 2 \)[/tex].
- The inequality [tex]\( y \geq -5x + 2 \)[/tex] means that the region of interest lies above or on the line [tex]\( y = -5x + 2 \)[/tex].
- Inequality 2: [tex]\( y < 3x - 1.5 \)[/tex]
- This translates to the boundary line [tex]\( y = 3x - 1.5 \)[/tex].
- The inequality [tex]\( y < 3x - 1.5 \)[/tex] means that the region of interest lies below the line [tex]\( y = 3x - 1.5 \)[/tex].
2. Find the point of intersection:
- To determine where these two lines intersect, set the equations equal to each other:
[tex]\( -5x + 2 = 3x - 1.5 \)[/tex]
- Combine like terms to solve for [tex]\( x \)[/tex]:
[tex]\[
\begin{align*}
-5x + 2 &= 3x - 1.5 \\
-5x - 3x &= -1.5 - 2 \\
-8x &= -3.5 \\
x &= \frac{-3.5}{-8} \\
x & = \frac{7}{16}
\end{align*}
\][/tex]
- Substitute [tex]\( x = \frac{7}{16} \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]:
[tex]\[
\begin{align*}
y &= -5 \left(\frac{7}{16}\right) + 2 \\
y &= -\frac{35}{16} + \frac{32}{16} \\
y &= -\frac{3}{16}
\end{align*}
\][/tex]
- The point of intersection is:
[tex]\[
\left( \frac{7}{16}, -\frac{3}{16} \right) \approx (0.4375, -0.1875)
\][/tex]
3. Graph the lines:
- Line 1: [tex]\( y = -5x + 2 \)[/tex]
- This line has a y-intercept at [tex]\( (0, 2) \)[/tex] and a slope of [tex]\(-5\)[/tex].
- The region of interest is above or on this line.
- Line 2: [tex]\( y = 3x - 1.5 \)[/tex]
- This line has a y-intercept at [tex]\( (0, -1.5) \)[/tex] and a slope of [tex]\(3\)[/tex].
- The region of interest is below this line.
4. Shade the regions:
- For [tex]\( y \geq -5x + 2 \)[/tex], shade the area above the line.
- For [tex]\( y < 3x - 1.5 \)[/tex], shade the area below the line.
- The solution region is where the two shaded areas overlap.
By following the described steps, the correct graph represents two boundary lines intersecting at the point [tex]\(\left( 0.4375, -0.1875 \right)\)[/tex], with the area of overlap satisfying both inequalities.
